[hackinthebochs]

Changing the base of number representation with a random basis feels like XORing a string with a random string, which is to say you're adding entropy equal to the random string. My thinking is that for any number representation M, you can get any other number representation N given a well-chosen base. So when presented with the encoded N, the original number could be any other number with the same number of digits. But once you put reasonable bounds on the base, you lose that flexibility and end up adding negligible entropy.

[GTP]

> So when presented with the encoded N, the original number could be any other number with the same number of digits

Not necessarily the same number of digits, when changing the base the number of digits may change as well. E.g., decimal 8 becomes 1000 in binary.